This paper presents a comprehensive theoretical analysis of the graph p-Laplacian regularized framelet network (pL-UFG) to establish a solid understanding of its properties. We conduct a convergence analysis on pL-UFG, addressing the gap in the understanding of its asymptotic behaviors. Further by investigating the generalized Dirichlet energy of pL-UFG, we demonstrate that the Dirichlet energy remains non-zero throughout convergence, ensuring the avoidance of over-smoothing issues. Additionally, we elucidate the energy dynamic perspective, highlighting the synergistic relationship between the implicit layer in pL-UFG and graph framelets. This synergy enhances the model's adaptability to both homophilic and heterophilic data. Notably, we reveal that pL-UFG can be interpreted as a generalized non-linear diffusion process, thereby bridging the gap between pL-UFG and differential equations on the graph. Importantly, these multifaceted analyses lead to unified conclusions that offer novel insights for understanding and implementing pL-UFG, as well as other graph neural network (GNN) models. Finally, based on our dynamic analysis, we propose two novel pL-UFG models with manually controlled energy dynamics. We demonstrate empirically and theoretically that our proposed models not only inherit the advantages of pL-UFG but also significantly reduce computational costs for training on large-scale graph datasets.

翻译：本文对图p-Laplacian正则化框架网络（pL-UFG）进行了全面的理论分析，旨在建立对其性质的深刻理解。我们对pL-UFG的收敛性展开分析，填补了对其渐近行为认知的空白。进一步通过研究pL-UFG的广义Dirichlet能量，我们证明该能量在收敛过程中始终非零，从而确保避免过平滑问题。此外，我们从能量动态视角阐明了pL-UFG中隐层与图框架之间的协同关系，这种协同增强了模型对同质与异质数据的适应性。值得注意的是，我们发现pL-UFG可被解释为广义非线性扩散过程，从而弥合了pL-UFG与图微分方程之间的鸿沟。重要的是，这些多角度分析得出一致结论，为理解与实现pL-UFG及其他图神经网络（GNN）模型提供了新颖洞见。最后，基于动态分析，我们提出了两种具有人工可控能量动态的新型pL-UFG模型。我们从经验与理论上证明，所提模型不仅继承了pL-UFG的优势，还显著降低了大规模图数据集上的训练计算成本。